Diffraction phases in atom interferometers

HAL Id: hal-00979356

https://hal.archives-ouvertes.fr/hal-00979356

Submitted on 15 Apr 2014

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

Diffraction phases in atom interferometers

Matthias Büchner, Rémi Delhuille, Alain Miffre, Cécile Robilliard, Jacques

Vigué, Caroline Champenois

To cite this version:

Matthias Büchner, Rémi Delhuille, Alain Miffre, Cécile Robilliard, Jacques Vigué, et al.. Diffraction

phases in atom interferometers. Physical Review A, American Physical Society, 2003, 68, pp.013607.

�10.1103/PhysRevA.68.013607�. �hal-00979356�

Diffraction phases in atom interferometers

M. Bu¨chner, R. Delhuille, A. Miffre, C. Robilliard, and J. Vigue´*

Laboratoire Collisions Agre´gats Re´activite´–IRSAMC, Universite´ Paul Sabatier and CNRS UMR 5589, 118, Route de Narbonne, 31062 Toulouse Cedex, France

C. Champenois

PIIM, Universite´ de Provence and CNRS UMR 6633, Centre de Saint Je´roˆme Case C21, 13397 Marseille Cedex 20, France ~Received 28 November 2002; published 16 July 2003!

Diffraction of atoms by lasers is a very important tool for matter wave optics. Although the process is well understood, the phase shifts induced by this diffraction process are not well known. In this paper, we make analytical calculations of these phase shifts in some simple cases and use these results to model the contrast interferometer recently built by Pritchard and co-workers. We thus show that the values of the diffraction phases are large and that they probably contribute to the phase noise observed in this experiment.

DOI: 10.1103/PhysRevA.68.013607 PACS number~s!: 03.75.Dg, 39.20.1q, 32.80.Pj

I. INTRODUCTION

In atom interferometry, laser diffraction is a very powerful and versatile tool ~for overviews, see Refs. @1,2#!. The dif-fraction of matter waves by a standing light wave was pro-posed by Kapitza and Dirac @3# in the case of electrons and generalized to atoms by Altshuler et al. @4#. Atom diffraction by light has been studied theoretically @5,6# and experimen-tally @7,8#, and this early work was followed by many studies too numerous to be quoted here. The phases of the diffraction amplitudes are rarely discussed in detail, with a few excep-tions like the work of Weitz et al. @9# and of Featonby et al.

@10#, in both cases for Raman adiabatic transfer, and the

work of Borde´ and co-workers @11,12#, which analyzes the general diffraction process in the rotating wave approxima-tion. Unfortunately, this approximation cannot be used for the elastic diffraction studied here.

In an interferometer, the diffraction phases modify the interference signals but this effect is difficult to detect, as it requires accurate phase measurements and it cancels in sym-metric interferometers, like the Mach-Zehnder interferom-eter. The goal of this paper is to present an analytical calcu-lation of diffraction phases in a simple case ~elastic diffraction by a laser standing wave! and to show the impor-tance of these diffraction phases in an existing experiment. We consider here diffraction in the Raman-Nath regime and second order Bragg diffraction in the weak field regime, and we apply these results to the contrast interferometer built by Gupta et al. @13#. The calculated diffraction phases are large in this interferometer, and as these phases depend sensitively on the laser power density used for the diffraction process, our calculation may explain the observed phase noise as re-sulting from fluctuations of this power density.

II. THE PROBLEM

We consider diffraction of slow ground state atoms by a near-resonant laser standing wave of frequency vL. For a

sufficiently large laser detuningd₅vL2v0, wherev0is the

resonance transition frequency, the probability of real excita-tion is negligible and the diffracexcita-tion process is coherent. In the dressed-atom picture @14#, the laser standing wave cre-ates a light shift potential V(x,t):

V_{~ x,t !5V0}~ t !cos2_{~ k}

Lx !

5V0₄~ t !@21exp~12ikLx !1exp~22ikLx !#, ~1!

where the envelope V0(t) is proportional to the laser power

density divided by the frequency detuning d, and kL is the

laser wave vector. We are going to forget the x-independent term, which simply shifts the energy zero and therefore has no effect, as long as all atoms experience the same potential. The motion along the y and z directions is free and will not be discussed. The natural energy unit is the atom recoil en-ergy \vrec5\2kL

2_{/2m, and we will measure the potential}

with this unit, by defining q(t) @15,16# as

q~ t !5V0~ t !/~ 4\vrec!. ~2!

Using a dimensionless timetdefined byt₅vrect, a

dimen-sionless spatial coordinate X5kLx, and a dimensionless

wave vector k_5kx/kL, the one-dimensional ~1D!

Schro¨-dinger equation becomes

i]C

]t 52

]2_C

]X21q~t!@exp~ 2iX !1exp~22iX!#C. ~3!

For a constant value of the potential q, the atom eigen-states are Bloch eigen-states @17,15,16#. Writing the Hamiltonian matrix corresponding to Eq. ~3! in the basis uk

&

of plane waves of momentum \k and using numerical diagonaliza-tion, we get the band structure «(k, p), with the pseudomo-mentum k belonging to the first Brillouin zone (21,k

<1) and the integer p labeling the bands @16#. Figure 1

presents the energy of the lowest Bloch states as a function ofkfor two values of the potential, q50 and q51, with two important features: when q is not equal to zero, band gaps *Electronic address: jacques.vigue@irsamc.ups-tlse.fr

appear at each crossing of the q50 folded parabola and energy shifts appear at the same time. These energy shifts are explained by perturbation theory: each free plane waveuk

&

is coupled to two other states uk62

&

, and the two coupling terms are equal. As the energy denominator is larger for the coupling to the upper state, all the levels are pushed upward

~except near the places where gaps open!, but the lowest

Bloch state is obviously pushed downward.

III. DIFFRACTION PHASES

In order to simplify the calculations, we consider that the atom is initially in a state of zero momentum, uc(t50)

&

5u0

&

. We first consider diffraction in the Raman-Nath re-gime. This approximation consists in neglecting the dynam-ics of the atom during the diffraction process produced by a pulse q(t) of duration tRN. This approximation is good

if the potential q(t) is intense, q@1, and if the pulse is brief, tRN!1. The validity range of this approximation is

given by @16,15#

tRN,1/~ 4

A

q ! ~4!

and the diffracted wave is a classic result:

uc~tRN!

&

5

(

p ~ 2i!

upu_J

upu~g!u2p

&

~5!

withg52qtRN. We have verified @16# that the Raman-Nath formula accurately predicts the diffraction probability of or-der 0 and 1, for finite values of the parameter q, as long as condition ~4! is satisfied, but we have not tested the phases of these diffraction amplitudes. They could be tested by using the diffraction amplitudes calculated @18# as a power series of 1/q.

Second order Bragg diffraction is due to the indirect cou-pling of theu62

&

free states, through theu0

&

state. As this coupling is a second order term in q, to make a consistent treatment, we must consider the five lowest-energy states, withk50,62,64. The Hamiltonian matrix has the

nonvan-ishing elements

^

2 puHu2p

&

54 p2 and

^

2 puHu2(p61)

&

5q.

Up to second order in q, the energy correction of theu0

&

state is E052q2/2, and the effective Hamiltonian coupling the

states u22

&

andu12

&

is

H_{e f f5}

F

41~q

2_/6!

~ q2/4!

~ q2_/4!

41~q2/6!

G

. ~6! We have tested the quality of this expansion limited to the q2 terms, by numerical diagonalization of the Hamiltonian ma-trix. The neglected terms ~in q4, etc.! are of the order of 1%

~10%! of the q2

terms if q50.3 (q51), thus giving an idea of the validity range of this calculation.

The dynamics is adiabatic if the potential q(t) varies slowly, but diffraction remains possible when two free states are degenerate, like theu62

&

states. The problem is equiva-lent to a Rabi oscillation exactly at resonance, for which an exact solution is available for any function q(t). For a pulse extending fromt1 tot2, the Rabi phasewr at the end of the

pulse is given by

wr5

E

t₁ t₂

~ q2_{/2!dt, ~7!}

and ifuc(t1)

&

5u62

&

the final state is

uc~t2!

&

5e$2i[4(t22t1)1(wr/3)]%

F

cos

S

wr

2

D

u62

&

2i sin

S

w₂r

D

u72

&

G

~8!

where the phase shift of the u62

&

states due to their mean energy shift has been expressed as a fraction of the Rabi phase. When uc(t1)

&

5u0

&

, the final state is the u0

&

state

with an extra phase shift, also due to its energy shift:

uc~t2!

&

5eiwru0

&

. ~9!

From now on, we consider awr5p pulse. If the wave

func-tion at timet1 is given by

uc~t1!

&

5

(

p522,0,12

ap~t1!up

&

, ~10!

the wave function at time t2 is given by

uc~t2!

&

5eipa0~t1!u0

&

1e[24i(t22t1)2(5ip/6)]@a22~t1!u12

&

1a12~t1!u22

&

]. ~11!

The phase factor exp_@24i(t22t1)# is due to the free

propa-gation of theu62

&

states and is not linked to the diffraction process. The interesting results are the diffraction phases equal to (1p) for theu0

&

state and (25p/6) for theu62

&

states. The opposite signs of the diffraction phases are a con-sequence of the opposite signs of the energy shifts of these levels. In the resulting phase difference, the level shift con-tribution, equal to 4p/3, is proportional to the Rabi phase

wr, taken equal top. In an experiment, the phase difference

FIG. 1. Plots of the energies « of the lowest Bloch states versus the pseudomomentumk: solid line, q51; dashed line, q50.

BU¨ CHNER et al. PHYSICAL REVIEW A 68, 013607 ~2003!

may differ from this calculated value, as a result of an imperfect p pulse or of other effects neglected here

~e.g., kÞ0).

IV. SIMPLE MODEL OF THE CONTRAST INTERFEROMETER OF GUPTA et al.

We now calculate the output signal of the contrast inter-ferometer developed by Gupta et al. @13#. This interferom-eter uses second order Bragg diffraction and Raman-Nath diffraction, and the atomic paths are represented in Fig. 2. The initial state is a Bose-Einstein condensate, approximated here by auk50

&

state. A first intense and brief pulse from

t50 to tRN is used to diffract this initial state in three co-herent statesu0

&

,u62

&

. Within the Raman-Nath approxima-tion, the wave function fortRN is given by

uc~tRN!

&

5J0u0

&

2iJ1@u12

&

1u22

&

], ~12!

the argument g of the Bessel functions being omitted for compactness. The best contrast @13# would be obtained with diffraction probabilities equal to 50% for the u0

&

state and 25% for each of theu62

&

states. It is impossible to perfectly satisfy these two conditions simultaneously, as the first one implies g51.13 whereas the second one implies g51.21.

We can nevertheless suppose that g'1.17. Although J2(1.17)'0.15, we will neglect here the second order

dif-fraction amplitudes, as was done in Ref. @13#. We assume that tRN is negligible so that free propagation starts att50

and lasts until the Bragg diffraction pulse, which extends fromt1tot2. Using Eq. ~11!, we get the wave function after

this pulse:

uc~t2!

&

5J0eipu0

&

1J1e24it2e24ip/3@u12

&

1u22

&

].

~13!

Free propagation goes on until a time t where the matter grating formed by the interference of these three states is read by the reflection of a laser beam. The atomic density as a function of X andt is deduced from the wave function:

z

^

Xuc~t!

&

z2 5J0 2 12J1 2 @11cos~4X!# 14J0J1cos~ 2X !cos

S

4t1 7p 3

D

. ~14!

The experimental signal S(t) is the intensity of the light reflected by this grating. This homodyne detection signal is proportional to the square of the cos(2X) modulation of the atomic density, with the following time dependence:

S~t!}cos2

S

_4t

17₃p

D

, ~15!

while the equation used by Gupta et al. is

S~t!}sin2_{~ 4t!. ~16!}

The difference between Eqs. ~15! and ~16! is important only if one wants to make an absolute prediction of the phase, but it has no consequence in the analysis carried out by Gupta

et al. @13#, because their fitted value ofvrec comes from the derivative of the phase with the time interval T @19#. How-ever, our result remains interesting as it may explain a large part of the observed phase noise, 200 mrad from shot to shot. In the 7p/3 phase of Eq. ~15!, 4p/3 is proportional to the Rabi phase, which is itself proportional to q2, i.e., to the square of the laser power density during the Bragg pulse. Therefore, a 1% variation of the laser power density changes the diffraction phase by 84 mrad.

Our calculation relies on several approximations, some of them being not very accurate in the experimental conditions of Gupta et al. @13#.

~i! The k50 approximation is an oversimplification but

the calculation withkÞ0 is more complex.

~ii! The first diffraction pulse used in the experiment is

1ms long, corresponding to tRN50.157. Assuming g

'1.17, we get q'3.7 and the validity condition ~4! requires

t<0.13. Therefore, the corrections to the Raman-Nath

phases are not fully negligible. We have also neglected the second order diffraction beams, which contribute to the signal.

~iii! As for the perturbation expansion used to describe

Bragg diffraction, the p pulse used is a Gaussian with a width of 7.6ms @13#. Assuming that q5qmaxexp@2(t

2T)2/(2st

2₎

#, withst53.8ms, i.e.,st'0.6, we get the value

q_max'2.4, well outside the validity range of our second

or-der perturbation expansion. Higher-oror-der terms in qn with

n54,6, . . . contribute to the phases and the sensitivity of the

diffraction phase to the laser power density may even be larger than predicted above.

Obviously, to describe this experiment very accurately, a full numerical modelization is needed and is feasible, as the problem reduces to a 1D Schro¨dinger equation, if atom-atom interactions are neglected. But, as noted by Gupta et al., the mean field effect of the condensate can also modify atomic propagation and this effect has not been considered here.

V. CONCLUSION

In this paper, we have made a simple and tutorial calcu-lation of the phase shifts of atomic waves due to the elastic diffraction process by a laser standing wave. We have calcu-lated the associated phase shift for the contrast interferometer of Pritchard and co-workers @13#, thus showing that it should FIG. 2. In the (x,t) plane, representation of the atomic paths

followed by the wave packets in the interferometer of Gupta et al.

@13#: Raman-Nath diffraction at time t50, second order Bragg

be possible to make an experimental test of the dependence of the diffraction phase shifts on potential strength and inter-action time. The present calculations are simple because of our assumptions: the Raman-Nath limit or perturbative re-gime, and vanishing initial momentum k50. An accurate

modelling of a real experiment requires numerical integra-tion of the Schro¨dinger equaintegra-tion to describe the diffracintegra-tion dynamics without any approximation.

We have considered only first and second order diffrac-tion. Higher diffraction orders up to order 8 have been ob-served @20–22# with moderate laser power densities. The leading term of the coupling matrix element responsible for diffraction order n behaves like qn _{@20#, whereas the leading}

terms of the energy shifts, responsible for the diffraction phase shifts, are always in q2. Therefore, for diffraction or-ders n.2, the control of the phase shifts will require a full knowledge of the pulse shape. For the second order of dif-fraction, the diffraction phase shifts and the Rabi phase are simply related, as long as second order perturbation theory is a good approximation.

We have made a systematic use of atomic Bloch states to describe atom diffraction by a laser, following our previous paper @16#. The introduction of Bloch states to describe at-oms in a laser standing waves is due to Letokhov and Mino-gin @23,24# in 1978 and also to Castin and Dalibard @25# in 1991. Their use is rapidly expanding, in particular to treat Bose-Einstein condensates in an optical lattice, as reviewed by Rolston and Phillips @26#. When coupled with reduced units as here, the atomic Bloch states represent a very effi-cient tool to get a simple understanding of the diffraction process.

ACKNOWLEDGMENTS

We thank C. Cohen-Tannoudji, J. Dalibard, and C. Salomon for very fruitful discussions, D. Pritchard for a very useful private communication, and Re´gion Midi Pyre´ne´es for financial support.

@1# Atom Interferometry, edited by P.R. Berman ~Academic, San Diego, 1997!.

@2# C.R. Acad. Sci., Ser IV: Phys., Astrophys. 2, 333 ~2001!, spe-cial issue on Bose-Einstein condensate and atom lasers, atom optics and interferometry, edited by A. Aspect and J. Dalibard. @3# P.L. Kapitza and P.A.M. Dirac, Proc. Cambridge Philos. Soc.

29, 297 ~1933!.

@4# S. Altshuler, L.M. Frantz, and R. Braunstein, Phys. Rev. Lett. 17, 231 ~1966!.

@5# R.J. Cook and A.F. Bernhardt, Phys. Rev. A 18, 2533 ~1978!. @6# A.F. Bernhardt and B.W. Shore, Phys. Rev. A 23, 1290 ~1981!. @7# E. Arimondo, H. Lew, and T. Oka, Phys. Rev. Lett. 43, 753

~1979!.

@8# P.E. Moskowitz et al., Phys. Rev. Lett. 51, 370 ~1983!. @9# M. Weitz, B.C. Young, and S. Chu, Phys. Rev. A 50, 2438

~1994!.

@10# P.D. Featonby et al., Phys. Rev. A 53, 373 ~1996!. @11# Ch.J. Borde´, in Atom Interferometry ~Ref. @1#!, p. 257. @12# Ch.J. Borde´ and C. La¨mmerzahl, Ann. Phys. ~Leipzig! 8, 83

~1999!.

@13# S. Gupta, K. Dieckmann, Z. Hadzibabic, and D.E. Pritchard,

Phys. Rev. Lett. 89, 140401 ~2002!.

@14# C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon Interactions ~Wiley, New York, 1998!.

@15# C. Keller et al., Appl. Phys. B: Lasers Opt. 69, 303 ~1999!. @16# C. Champenois et al., Eur. Phys. J. D 13, 271 ~2001!. @17# M. Horne, I. Jex, and A. Zeilinger, Phys. Rev. A 59, 2190

~1999!.

@18# M.V. Berry, The Diffraction of Light by Ultrasound ~Aca-demic, London, 1966!.

@19# D. Pritchard ~private communication!.

@20# D.M. Giltner, R.W. McGowan, and Siu Au Lee, Phys. Rev. A 52, 3966 ~1995!.

@21# D.M. Giltner, R.W. McGowan, and Siu Au Lee, Phys. Rev. Lett. 75, 2638 ~1995!.

@22# A.E.A. Koolen et al., Phys. Rev. A 65, 041601~R! ~2002!. @23# V.S. Letokhov and V.G. Minogin, Zh. Eksp. Teor. Fiz. 74, 1318

~1978!.

@24# V.S. Letokhov and V.G. Minogin, Phys. Rep. 73, 1 ~1981!. @25# Y. Castin and J. Dalibard, Europhys. Lett. 14, 761 ~1991!. @26# S.L. Rolston and W.D. Phillips, Nature ~London! 416, 219

~2002!.

BU¨ CHNER et al. PHYSICAL REVIEW A 68, 013607 ~2003!